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Abstract

We study the existence and multiplicity of positive solutions for a system of nonlinear
Riemann-Liouville fractional differential equations, subject to integral boundary
conditions. The nonsingular and singular cases for the nonlinearities are investigated.

Keywords:

1 Introduction

We consider the system of nonlinear ordinary fractional differential equations

with the integral boundary conditions

where , , and denote the Riemann-Liouville derivatives of orders α and β, respectively, and the integrals from (BC) are Riemann-Stieltjes integrals.

Under sufficient conditions on functions f and g, which can be nonsingular or singular in the points and/or , we study the existence and multiplicity of positive solutions of problem (S)-(BC).
We use the Guo-Krasnosel’skii fixed point theorem (see [1]) and some theorems from the fixed point index theory (from [2] and [3]). By a positive solution of problem (S)-(BC) we mean a pair of functions satisfying (S) and (BC) with , for all and , . The system (S) with , and the boundary conditions (BC) where H and K are scale functions (that is, multi-point boundary conditions) has been investigated
in [4] (the nonsingular case) and [5] (the singular case). In [6], the authors give sufficient conditions for λ, μ, f, and g such that the system

with the boundary conditions (BC) with H and K scale functions, has positive solutions (, for all , and ).

Fractional differential equations describe many phenomena in various fields of engineering
and scientific disciplines such as physics, biophysics, chemistry, biology, economics,
control theory, signal and image processing, aerodynamics, viscoelasticity, electromagnetics,
and so on (see [7-13]).

In Section 2, we present the necessary definitions and properties from the fractional
calculus theory and some auxiliary results dealing with a nonlocal boundary value
problem for fractional differential equations. In Section 3, we give some existence
and multiplicity results for positive solutions with respect to a cone for our problem
(S)-(BC), where f and g are nonsingular functions. The case when f and g are singular at and/or is studied in Section 4. Finally, in Section 5, we present two examples which illustrate
our main results.

2 Preliminaries and auxiliary results

We present here the definitions, some lemmas from the theory of fractional calculus
and some auxiliary results that will be used to prove our main theorems.

Definition 2.1 The (left-sided) fractional integral of order of a function is given by

provided the right-hand side is pointwise defined on , where is the Euler gamma function defined by , .

Definition 2.2 The Riemann-Liouville fractional derivative of order for a function is given by

where , provided that the right-hand side is pointwise defined on .

The notation stands for the largest integer not greater than α. We also denote the Riemann-Liouville fractional derivative of f by . If then for , and if then for .

Proof The first inequality (a) is evident. For part (b), for and , , we deduce

Therefore, we obtain the inequalities (b) of this lemma. □

Lemma 2.8Assume thatis a nondecreasing function and, , and, for all. Then the solution, of problem (1)-(2) satisfies the inequality.

Proof For , , , we have

Then we deduce the conclusion of this lemma. □

We can also formulate similar results as Lemmas 2.3-2.8 above for the fractional differential
equation

(6)

with the integral boundary conditions

(7)

where , , is a nondecreasing function and . We denote by , , , , , and the corresponding constants and functions for the problem (6)-(7) defined in a similar
manner as , , , , , and , respectively.

3 The nonsingular case

In this section, we investigate the existence and multiplicity of positive solutions
for our problem (S)-(BC) under various assumptions on nonsingular functions f and g.

We present the basic assumptions that we shall use in the sequel.

(H1) are nondecreasing functions, , .

(H2) The functions are continuous and for all .

A pair of functions is a solution for our problem (S)-(BC) if and only if is a solution for the nonlinear integral system

We consider the Banach space with supremum norm and define the cone by .

We also define the operators by

and , by

Under the assumptions (H1) and (H2), using also Lemma 2.6, it is easy to see that
, ℬ, and are completely continuous from P to P. Thus the existence and multiplicity of positive solutions of the system (S)-(BC)
are equivalent to the existence and multiplicity of fixed points of the operator .

Theorem 3.1Assume that (H1)-(H2) hold. If the functionsfandgalso satisfy the conditions:

(H3) There exist positive constantsandsuch that

(H4) There exists a positive constantsuch that

then the problem (S)-(BC) has at least one positive solution, .

Proof Because the proof of the theorem is similar to that of Theorem 3.1 from [4], we will sketch some parts of it. From assumption (i) of (H3), we deduce that there
exist such that

(8)

Then for , by using (8), Lemma 2.6, and Lemma 2.7, we obtain after some computations

(9)

where .

For c given in (H3), we define the cone , where . From our assumptions and Lemma 2.8, for any , we can easily show that and , that is, and .

We now consider the function , , with for all . We define the set

We will show that and M is a bounded subset of X. If , then there exists such that , . From the definition of , we have

where is defined by . Therefore, , and from the definition of , we get

(10)

From (ii) of assumption (H3), we conclude that for there exists such that

(11)

where , .

For and , by using Lemma 2.7 and the relations (9) and (11), it follows that

where .

Hence, , and so

(12)

Now from relations (10) and (12), one obtains , for all , that is, M is a bounded subset of X.

Besides, there exists a sufficiently large such that

From [2], we deduce that the fixed point index of the operator over with respect to P is

(13)

Next, from assumption (H4), we conclude that there exist and such that

(14)

where , , . Hence, for any and , we obtain

(15)

Therefore, by (14) and (15), we deduce that for any and

This implies that for all . From [2], we conclude that the fixed point index of the operator over with respect to P is

(16)

Combining (13) and (16), we obtain

We deduce that has at least one fixed point , that is, .

Let . Then is a solution of (S)-(BC). In addition . Indeed, if we suppose that , for all , then by using (H2) we have , for all . This implies , for all , which contradicts . The proof of Theorem 3.1 is completed. □

Using similar arguments as those used in the proofs of Theorem 3.2 and Theorem 3.3
in [4], we also obtain the following results for our problem (S)-(BC).

Theorem 3.2Assume that (H1)-(H2) hold. If the functionsfandgalso satisfy the conditions:

(H5) There exists a positive constantsuch that

(H6) There existssuch that

then the problem (S)-(BC) has at least one positive solution, .

Theorem 3.3Assume that (H1)-(H3), and (H6) hold. If the functionsfandgalso satisfy the condition:

(H7) For each, andare nondecreasing with respect tou, and there exists a constantsuch that

where, , , and, are defined in Section 2, then the problem (S)-(BC) has at least two positive solutions, , .

4 The singular case

In this section, we investigate the existence of positive solutions for our problem
(S)-(BC) under various assumptions on functions f and g which may be singular at and/or .

The basic assumptions used here are the following.

() ≡ (H1).

() The functions and there exist , , , with , , , such that

We consider the Banach space with supremum norm and define the cone by . We also define the operator by

Lemma 4.1Assume that ()-() hold. Thenis completely continuous.

Proof We denote by and . Using (), we deduce that and . By Lemma 2.6 and the corresponding lemma for , we see that maps P into P.

We shall prove that maps bounded sets into relatively compact sets. Suppose is an arbitrary bounded set. Then there exists such that for all . By using () and Lemma 2.7, we obtain for all , where , and . In what follows, we shall prove that is equicontinuous. By using Lemma 2.4, we have

Therefore, for any , we obtain

So, for any , we deduce

(17)

We denote

For the integral of the function h, by exchanging the order of integration, we obtain

For the integral of the function μ, we have

(18)

We deduce that . Thus for any given with and , by (17), we conclude

(19)

From (18), (19), and the absolute continuity of the integral function, we find that
is equicontinuous. By the Ascoli-Arzelà theorem, we deduce that is relatively compact. Therefore is a compact operator. Besides, we can easily show that is continuous on P. Hence is completely continuous. □

Theorem 4.1Assume that ()-() hold. If the functionsfandgalso satisfy the conditions:

() There existwithsuch that

() There existwithandsuch that

then the problem (S)-(BC) has at least one positive solution, .

Proof Because the proof of this theorem is similar to that of Theorem 3 in [5], we will sketch some parts of it. For c given in (), we consider the cone , where . Under assumptions ()-(), we obtain . By (), we deduce that there exist and such that

(20)

By using (20) and (), for any , we conclude

By the definition of , we can choose sufficiently large such that

(21)

From (), we deduce that there exist positive constants , , and such that

(22)

where and . From the assumption and the continuity of , we conclude that there exists sufficiently small such that for all , where . Therefore for any and , we have

(23)

By (22), (23), Lemma 2.7, and Lemma 2.8, for any and , we obtain

Therefore

(24)

By (21), (24), and the Guo-Krasnosel’skii fixed point theorem, we deduce that has at least one fixed point . Then our problem (S)-(BC) has at least one positive solution where . The proof of Theorem 4.1 is completed. □

Using similar arguments as those used in the proof of Theorem 2 in [5] (see also [14] for a particular case of the problem studied in [5]), we also obtain the following result for our problem (S)-(BC).

Theorem 4.2Assume that ()-() hold. If the functionsfandgalso satisfy the conditions:

() There existwithsuch that

() There existwithandsuch that

then the problem (S)-(BC) has at least one positive solution, .

5 Examples

Let (), (),

and for all . Then and .

We consider the system of fractional differential equations

with the boundary conditions

Then we obtain , . We also deduce

and for all .

For the functions and , we obtain

and

Example 1 We consider the functions

where , , , , . We have , . Then . The functions and are nondecreasing with respect to u, for any , and for and the assumptions (H3) and (H6) are satisfied; indeed we obtain

We take and then and . If , then the assumption (H7) is satisfied. For example, if , , , and (e.g.), then the above inequality is satisfied. By Theorem 3.3, we deduce that the problem
()-() has at least two positive solutions.

Example 2 We consider the functions

with and . Here and , where

We have , .

In (), for , and , we obtain

In (), for , , , and , we have

For example, if , , , , , , the above conditions are satisfied. Then, by Theorem 4.2, we deduce that the problem
()-() has at least one positive solution.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to this paper. Both authors read and approved the
final manuscript.

Acknowledgements

The work of R Luca was supported by the CNCS grant PN-II-ID-PCE-2011-3-0557, Romania.